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Relationship between zeros and coefficients of a cubic polynomial

If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

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Student-friendly explanation

This result extends the quadratic relationship to cubics. It is useful when one or more zeros are known, or when a cubic polynomial is formed from given roots. The signs must be read carefully from the standard form. This relation saves time and is often used in higher-level algebra questions.

How to write this in exams

  1. 1

    Start with the exact idea

    If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

  2. 2

    Then show how to use it

    1. Compare the polynomial with ax^3+bx^2+cx+d. 2. Note a, b, c, and d with signs. 3. Apply the three formulas carefully. 4. Use them to verify or construct the polynomial.

  3. 3

    Add one concrete example

    For x^3-6x^2+11x-6, sum of zeros is 6, sum of pairwise products is 11, and product of zeros is 6.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to write the product as d/a. The negative sign is required because the cubic relation changes sign for the constant term.

Definition

If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

Example

For x^3-6x^2+11x-6, sum of zeros is 6, sum of pairwise products is 11, and product of zeros is 6.

Rule to remember

For zeros α, β, γ of ax^3+bx^2+cx+d: α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

Memory hook

Cubic: sum, pair-sum, product; watch the signs.

Examples and method

Worked example

For x^3-3x^2-4x+12, the product of zeros is -12/1=-12. If the zeros are 2, -2, and 3, their product is also -12.

Method to apply

1. Compare the polynomial with ax^3+bx^2+cx+d. 2. Note a, b, c, and d with signs. 3. Apply the three formulas carefully. 4. Use them to verify or construct the polynomial.

Diagram support

A cubic graph can cross the x-axis up to three times, and those intercepts correspond to the zeros used in these relations.

How CBSE asks it

Find one relation from the coefficients, verify a set of roots, or construct a cubic from zeros.

Avoid common mistakes

Common confusion

Students often forget that the product of three zeros has a negative sign with d/a in the standard form ax^3+bx^2+cx+d.

Common wrong answer

A common wrong answer is to write the product as d/a. The negative sign is required because the cubic relation changes sign for the constant term.

Exam tip

Write the cubic in standard form first, then apply all three relations one by one.

Quick check

What is the product of zeros of ax^3+bx^2+cx+d?

The product of the zeros is -d/a. The negative sign is important, so students should read the constant term carefully from the standard form.

Answer writing and exam use

1-mark answer

If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

2-mark answer

If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a. For zeros α, β, γ of ax^3+bx^2+cx+d: α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a. For x^3-6x^2+11x-6, sum of zeros is 6, sum of pairwise products is 11, and product of zeros is 6.

3-mark answer

This result extends the quadratic relationship to cubics. It is useful when one or more zeros are known, or when a cubic polynomial is formed from given roots. The signs must be read carefully from the standard form. This relation saves time and is often used in higher-level algebra questions. For zeros α, β, γ of ax^3+bx^2+cx+d: α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a. For x^3-3x^2-4x+12, the product of zeros is -12/1=-12. If the zeros are 2, -2, and 3, their product is also -12. Find one relation from the coefficients, verify a set of roots, or construct a cubic from zeros. A common wrong answer is to write the product as d/a. The negative sign is required because the cubic relation changes sign for the constant term.
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